Questions about the branch of abstract algebra that deals with groups.

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13
votes
1answer
193 views

Free generators for the fat commutator subgroup

There is a homomorphism $\langle x,y\rangle\to\langle x\rangle$ of free groups, sending $y$ to $1$. We can combine this with the other obvious homomorphism to get a surjective homomorphism $$ ...
1
vote
0answers
68 views

associativity of the extension of finie groups [closed]

Following my previous question I have two questions: 1.the extensions of the group is associative or not, i.e. as we know by the notations of atlas if $S={\rm PSL}(2,q)$, where $q=p^f>9$, then ...
7
votes
0answers
95 views

Number of matrices with 'small' entries in integer matrix groups

Let $n$ be a positive integer, and let $G \leq {\rm GL}(n,\mathbb{Z})$. Given a bound $b \in \mathbb{N}$, let $e_b$ be the number of elements of $G$ all of whose matrix entries have absolute value ...
3
votes
0answers
88 views

Profinite topology on free metabelian group

Let $M$ be free metabelian group of rank $n$. By work of Coulbois, M is $LERF$ and is not $RZ_2$, that means every finitely generated subgroup of $M$ is closed in the profinite topology of $M$ but the ...
7
votes
1answer
174 views

Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which 1- is finitely generated by $S$, 2- does not have property (T), 3- admits infinitely many finite quotients which do not factor through an homomorphism $G ...
4
votes
1answer
137 views

Extensions of $\Bbb Z_3$ by $PGL(2,q)$ where $q$ is odd

Let $q$ be odd. If $G$ is a finite group such that $G$ has a normal subgroup $H$ of order $3$ such that $G/H\cong {\rm PGL}(2,q)$, what can we say about $G$. Is it true in general that $G\cong {\Bbb ...
7
votes
0answers
166 views

A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that: For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$ contains infinitely many finite index ...
5
votes
2answers
309 views

Inverse Galois problem for simple Lie type groups

Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...
4
votes
0answers
107 views

Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices. The (conjugate) Steinberg idempotent is defined to be ...
5
votes
1answer
193 views

To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...
2
votes
1answer
116 views

k-fellow traveler property and automatic structur

Let P be a permutation group with some generating set S and let W be the word acceptor automaton of P, if I know the value of k (k-fellow-traveller property of CayleyGraph CG(P,S)). I realized that ...
0
votes
0answers
84 views

Abelian centralizer groups (CA-groups)

I am searching for all information about CA-groups [abelian centralizer groups] and i just found a German book [Huppert] and Nilpotent Centralizer group of Suzuki in 44 pages and Group theory book of ...
3
votes
0answers
93 views

Schur Multiplier of Tarski Monsters

Is it known whether the Schur Multiplier of the Tarski monsters are finitely generated?
12
votes
0answers
217 views

A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
3
votes
2answers
132 views

On the Magnus Representation of Free Metabelian Group

Let $F=\langle x_1,x_2\rangle$ be a free group of rank $2$ and $\Phi=F/F''=\langle \overline{x}_1, \overline{x}_2\rangle$ where $F''$ is second derived subgroup of $F$ (i.e. $F'=[F,F]$ and ...
7
votes
2answers
390 views

asymptotic for the number of involutions in GL(n,2)

Is it known how the number of involutions in $GL_n(2)$, the group of $n\times n$ matrices over $\mathbb{Z}/2\mathbb{Z}$, behaves as $n\to\infty$ ? Equivalently, one may ask this for the number of ...
2
votes
1answer
75 views

The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime? Let $p_1, \dots, p_r$ be $r$ distinct odd primes. Set $$G = ...
7
votes
1answer
104 views

What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$. What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$? Here we denote by $d(G)$ the smallest ...
11
votes
1answer
288 views

Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question. In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...
5
votes
1answer
182 views

Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements. Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ? I am interested in ...
9
votes
1answer
176 views

Inducing up the group homomorphism between mapping class groups

There are many ways to embed the braid group into the mapping class group of a surface. To describe one of them, let ${C}_{2g+2}(\mathbb{D}^2)$ be the configuration of unordered $2g+2$ points in the ...
14
votes
2answers
588 views

Braid group on 4 strands

Consider the braid group $B_4$ with generators $a=\ $, $b=\ $ and $c=\ $ Assume $$\alpha, \alpha' \in \langle a,c\rangle{\smallsetminus}(\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle ...
2
votes
0answers
134 views

calculation in a group ring

I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...
4
votes
1answer
139 views

Centralizer of hermitian matrices with zero trace

In Quantum Physics one often has to deal with commutators. Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero! One can easily relate it to ...
5
votes
0answers
72 views

Normalizer of a reflection supgroup of a finite complex reflection group

Let $W \leq \operatorname{GL}(\mathbb{C}^\ell)$ be a finite complex reflection group acting on the finite dimensional complex vectorspace $\mathbb{C}^\ell$. Let $W' \leq W$ be a reflection subgroup, ...
4
votes
0answers
272 views

Characterization for special linear group over finite fields

Thanks for any help or comments. In my research I need a characterization for special linear groups over finite fields by some information of its subgroups, especially centralizers. I saw the ...
4
votes
1answer
95 views

Computations with conetypes of hyperbolic groups

I'd like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group ...
0
votes
0answers
46 views

The name of a class of linearly ordered groups

My friend asked me to ask his question here. Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that ...
4
votes
0answers
142 views

Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?

Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...
3
votes
1answer
117 views

extending homomorphisms to semidirect products

Let $A = F_2$ be the free group on two generators (not sure if this is important.) Suppose you have a semidirect product $A\rtimes C$ coming from some homomorphism $\beta: C\rightarrow ...
7
votes
4answers
582 views

fixed points of permutation groups

As is well-known (see, for example, a nice exposition by our own Qiaochu: https://qchu.wordpress.com/2012/11/07/fixed-points-of-random-permutations/) that the distribution of the number of fixed ...
5
votes
0answers
82 views

Maximum relator and hyperbolicity

It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows: Suppose ...
3
votes
0answers
122 views

Nilpotent Groups of Generalized Prime Exponent

A group $G$ is called of generalized exponent $n$ if there exists elements $a_1,\dots,a_n \in G$ such that $x^{a_1}\cdots x^{a_n}=1$ for all $x\in G$, where $x^a=a^{-1}xa$. See the following question ...
5
votes
1answer
133 views

Minimal number of algebra generators of a group ring

Let $k$ be a commutative ring with unit and let $(A,\varepsilon)$ be a (not necessarily commutative) augmented, finitely generated $k$-algebra with augmentation ideal $I$. If $\mu(A)$ denotes the ...
4
votes
0answers
77 views

Sharp isoperimetry in the discrete Heisenberg group

The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case? More ...
9
votes
0answers
176 views

Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups. Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ...
5
votes
1answer
209 views

Is there a one relator group with property (T)?

Is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup generated by $x$ has Kazhdan's property $\mathrm{(T)}$ ?
3
votes
1answer
266 views

Maximal ideal of group ring

Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...
2
votes
0answers
60 views

Actions on spaces with measured walls

In geometric group theory, the question of whether or not a group acts nicely on a CAT(0) cube complex, or equivalently on a median graph, is of interest. The same question for actions on spaces with ...
5
votes
1answer
173 views

Do one-relator groups satisfy Haagerup property?

The question is in the title: Do one-relator groups satisfy Haagerup property? I think the answer is known at least in some specific cases, but is the problem completely solved?
9
votes
1answer
254 views

Is the free product $\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}$ linear over $\mathbb{Z}$?

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I want to know if $H$ is a ($\mathbb{Z}$)linear group that is to say is there an injective homomorphism $f: H\to GL_m(\mathbb{Z})$ ...
72
votes
2answers
3k views

$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
16
votes
0answers
517 views

How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
4
votes
0answers
72 views

Methods for showing homology of a subgroup survives to the larger group

Suppose we have an inclusion of groups $G_1<G_2$. I am curious about what methods there are out there for analyzing the map $H_k(G_1;\mathbb Q)\to H_k(G_2;\mathbb Q)$. In particular, what are tools ...
4
votes
0answers
82 views

Smooth admissible representations, Hom, tensor and extension of scalars

(Remark: This has previously been posted on math.stackexchange, but I believe it might be suitable for this site as well. ...
4
votes
2answers
384 views

What does it mean to take the diagonal of the group $SU(2) \times SU(2) $?

I am reading Witten's paper on topological field theories, in specific the topological twist in page 359. In order to perform the twist he takes the diagonal subgroup of $K = SU(2)_{\text{Right}} ...
7
votes
2answers
320 views

Theorems of the Galois groups of quintics appears not to work for the ${F}_{20}$ group determination

I am computing the Galois groups of quintics using the theorems from Ryan Kavanagh paper "On Irreducible Rational Quintics" using the decic resolvent ${P}_{10} \left({x}\right) = \prod\limits_{1 \le i ...
4
votes
2answers
315 views

Explicit description of the principal block of the symmetric group

Let $k$ be a field of prime characteristic $p$ and $\Sigma_n$ be the symmetric group. If I have a concrete $k[\Sigma_n]$-module $M,$ how to compute the direct summand corresponding to the principal ...
3
votes
1answer
162 views

Is there a bound on the rank of finite index subgroup of SL_3(Z)?

Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?
4
votes
1answer
98 views

Is being Noetherian a quasi-isometric invariance for f.g. groups?

Recall that a group $G$ satisfies max (or is said to be Noetherian) if all its proper subgroups are finitely generated. Similarly $G$ satiesfies max-n if all its normal subgroups are normal closures ...